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Mimetic Gravity Description of Loop Quantum Cosmology: Reinterpreting the Bounce and Inflation from a Curvature Perspective

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Mimetic Gravity Description of Loop Quantum Cosmology: Reinterpreting the Bounce and Inflation from a Curvature Perspective sid.inpe.br/mtc-m21c/2019/04.17.16.57-TDI MIMETIC GRAVITY DESCRIPTION OF LOOP QUANTUM COSMOLOGY: REINTERPRETING THE BOUNCE AND INFLATION FROM A CURVATURE PERSPECTIVE Eunice Valtânia de Jesus Bezerra Doctorate Thesis of the Graduate Course in Astrophysics, guided by Dr. Oswaldo Duarte Miranda, approved in April 29, 2019. URL of the original document: <http://urlib.net/8JMKD3MGP3W34R/3T62Q42> INPE São José dos Campos 2019 PUBLISHED BY: Instituto Nacional de Pesquisas Espaciais - INPE Gabinete do Diretor (GBDIR) Serviço de Informação e Documentação (SESID) CEP 12.227-010 São José dos Campos - SP - Brasil Tel.:(012) 3208-6923/7348 E-mail: [email protected] BOARD OF PUBLISHING AND PRESERVATION OF INPE INTELLECTUAL PRODUCTION - CEPPII (PORTARIA No 176/2018/SEI-INPE): Chairperson: Dra. Marley Cavalcante de Lima Moscati - Centro de Previsão de Tempo e Estudos Climáticos (CGCPT) Members: Dra. Carina Barros Mello - Coordenação de Laboratórios Associados (COCTE) Dr. Alisson Dal Lago - Coordenação-Geral de Ciências Espaciais e Atmosféricas (CGCEA) Dr. Evandro Albiach Branco - Centro de Ciência do Sistema Terrestre (COCST) Dr. Evandro Marconi Rocco - Coordenação-Geral de Engenharia e Tecnologia Espacial (CGETE) Dr. Hermann Johann Heinrich Kux - Coordenação-Geral de Observação da Terra (CGOBT) Dra. Ieda Del Arco Sanches - Conselho de Pós-Graduação - (CPG) Silvia Castro Marcelino - Serviço de Informação e Documentação (SESID) DIGITAL LIBRARY: Dr. Gerald Jean Francis Banon Clayton Martins Pereira - Serviço de Informação e Documentação (SESID) DOCUMENT REVIEW: Simone Angélica Del Ducca Barbedo - Serviço de Informação e Documentação (SESID) André Luis Dias Fernandes - Serviço de Informação e Documentação (SESID) ELECTRONIC EDITING: Ivone Martins - Serviço de Informação e Documentação (SESID) Cauê Silva Fróes - Serviço de Informação e Documentação (SESID) sid.inpe.br/mtc-m21c/2019/04.17.16.57-TDI MIMETIC GRAVITY DESCRIPTION OF LOOP QUANTUM COSMOLOGY: REINTERPRETING THE BOUNCE AND INFLATION FROM A CURVATURE PERSPECTIVE Eunice Valtânia de Jesus Bezerra Doctorate Thesis of the Graduate Course in Astrophysics, guided by Dr. Oswaldo Duarte Miranda, approved in April 29, 2019. URL of the original document: <http://urlib.net/8JMKD3MGP3W34R/3T62Q42> INPE São José dos Campos 2019 Cataloging in Publication Data Bezerra, Eunice Valtânia de Jesus. B469m Mimetic gravity description of loop quantum cosmology: reinterpreting the bounce and inflation from a curvature perspective / Eunice Valtânia de Jesus Bezerra. – São José dos Campos : INPE, 2019. xxii + 128 p. ; (sid.inpe.br/mtc-m21c/2019/04.17.16.57-TDI) Thesis (Doctorate in Astrophysics) – Instituto Nacional de Pesquisas Espaciais, São José dos Campos, 2019. Guiding : Dr. Oswaldo Duarte Miranda. 1. Mimetic Gravity. 2. Loop Quantum Cosmology. 3. Higgs inflation. 4. Curvature. 5. Primordial universe. I.Title. CDU 550.312:524.8 Esta obra foi licenciada sob uma Licença Creative Commons Atribuição-NãoComercial 3.0 Não Adaptada. This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License. ii To my mother Virgulina, for all her love and support. v ACKNOWLEDGEMENTS I would like to thank my mother Virgulina for all her love and effort to provide me with the best opportunities she could. I am extremely proud to be called her daughter. I would like to thank my advisor Dr. Oswaldo Duarte Miranda for the honor and privilege to work with him. I could not have asked for a better person to guide me during my PhD. I am grateful for the support of my blood family, including my father, sister, grand- mother, aunts, cousins, uncles and so on. I am really thankful for the heart family I found at INPE, in particular, Lorena, Carol, Aysses, Isabel, Dinelsa, Fabrícia, Eduardo, Manuel and Fernando. I would like to thank my teachers for all the teachings I learned along these years. I thank the INPE staff for the help and kindness I was treated since the start of my master course. I would like to thank CAPES for the graduate research fellowship. vii ABSTRACT The physics of the early universe remains a mystery to cosmologists. The primordial universe is within energy scales in which the quantum gravity effects may play a significant role. A consistent quantum gravity theory would be composed with the best elements from general relativity and quantum field theory. In our work, we focus on Loop Quantum Gravity (LQG) or being more specific, we are interested in its cosmological counterpart, called Loop Quantum Cosmology (LQC), at the effective level. The results coming from Effective LQC are promising, mainly, the replacement of the initial singularity by a bounce. On the other hand, there is a very powerful approach known as Mimetic Gravity (MG) that can also produce a bounce. The original theory was not built with this purpose, but it was generalized and became capable of reproducing different cosmological evolutionary scenarios. Above all, Langlois et al.(2017) constructed a MG description of LQC in which the Effective LQC dynamics was reproduced. In particular, we explore the treatment employed to obtain the cosmological evolution in a curved Friedmann-Robertson- Walker space-time and the curvature potential presented in it. We build our own formulation reinterpreting the curvature potential role, at the same time, we empha- size its direct relation with matter. After, we apply the developed formalism to Higgs inflationary scenario, taking advantage of the intrinsic relation between the Higgs field and curvature. Moreover, we discuss the primordial gravitational waves and its essential role as fundamental proofs to validate any cosmological model whose aim is to describe the primordial universe dynamics. Keywords: Mimetic Gravity. Loop Quantum Cosmology. Higgs Inflation. Curvature. Primordial universe. ix DESCRIÇÃO DA COSMOLOGIA QUÂNTICA EM LAÇOS PELA GRAVIDADE MIMÉTICA: REINTERPRETANDO O RICOCHETE E A INFLAÇÃO PELA PERSPECTIVA DA CURVATURA RESUMO A física do universo primordial permanece um mistério para os cosmólogos. O uni- verso primordial está dentro do intervalo de energia no qual os efeitos gravitacionais quânticos desempenhariam um papel significativo. Uma teoria de gravitação quân- tica consistente seria composta pelos melhores elementos da relatividade geral e teo- ria quântica de campo. No nosso trabalho, nós focamos na Gravitação Quântica em Laços (GQL), ou sendo mais específicos, estamos interessados na sua contrapartida cosmológica, denominada Cosmologia Quântica em Laços (CQL), no nível efetivo. Os resultados oriundos da CQL são promissores, principalmente, a substituição da singularidade inicial por um ricochete. Por outro lado, existe uma poderosa abor- dagem conhecida como Gravidade Mimética (GM) que também pode produzir um ricochete. A teoria original não foi construída com esse propósito, mas foi general- izada, tornando-se capaz de reproduzir diferentes cenários de evolução cosmológica. Sobretudo, Langlois et al.(2017) construiu uma descrição da CQL através da GM, na qual a dinâmica efetiva da CQL pode ser reproduzida. Em particular, exploramos o tratamento empregado para obter a evolução cosmológica de um espaço-tempo Friedmann-Robertson-Walker curvo, além do potencial de curvatura nele apresen- tado. Construímos nossa formulação própria reinterpretando o papel do potencial de curvatura, ao mesmo tempo em que enfatizamos sua relação direta com a matéria. A seguir, aplicamos nossa formulação para o cenário da inflação de Higgs, usufruindo da relação intrínseca entre o campo de Higgs e a curvatura. Além disso, discutimos as ondas gravitacionais primordiais, bem como seu papel essencial como uma prova fundamental utilizada para confirmar a validade de um modelo cosmológico cujo objetivo é descrever a dinâmica do universo primordial. Palavras-chave: Gravidade Mimética. Cosmologia Quântica em Laços. Inflação de Higgs. Curvatura. Universo primordial. xi LIST OF FIGURES Page 1.1 Standard universe evolution timeline .................... 4 1.2 Horizon problem ............................... 11 1.3 Solution to the horizon problem....................... 14 1.4 Particle horizon and Hubble radius..................... 14 1.5 Universe evolution .............................. 22 2.1 Space-time foliation.............................. 27 2.2 Relation between the spacelike hypersurfaces in the universe foliation . 27 2.3 Extrinsic curvature of the hypersurface................... 30 2.4 Spin network ................................. 45 2.5 Possible previous configuration of our universe............... 49 2.6 Scheme evolution of the very early universe ................ 51 4.1 Solutions of the tracking equation...................... 78 4.2 Evolution of the potential terms regarding MG and HI .......... 79 4.3 Evolution of the modified Friedmann equation............... 87 5.1 BICEP2 CMB power spectrum for B-mode................. 92 5.2 CMB power spectrum ............................ 93 5.3 CMB spectrum for a bounce universe.................... 94 5.4 CMB spectrum from Loop Quantum Cosmology.............. 95 5.5 CMB spectrum from Massive Gravity.................... 96 6.1 MG as the bridge between LQC and HI ..................101 C.1 Cosmic Microwave Background .......................125 C.2 CMB polarization patterns..........................128 xiii LIST OF TABLES Page 1.1 Epochs of the standard universe evolution ................. 5 2.1 Variables that plays similar roles ...................... 32 2.2 Different pairs of variables from Loop Quantum Cosmology . 44 2.3 Evolutionary stages of the primordial universe in Loop
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